Properties

Label 8-825e4-1.1-c1e4-0-6
Degree $8$
Conductor $463250390625$
Sign $1$
Analytic cond. $1883.32$
Root an. cond. $2.56664$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s − 2·6-s + 8·7-s − 5·8-s + 4·11-s + 2·12-s + 4·13-s − 16·14-s + 5·16-s − 2·17-s − 5·19-s + 8·21-s − 8·22-s + 14·23-s − 5·24-s − 8·26-s + 16·28-s − 5·29-s − 7·31-s + 2·32-s + 4·33-s + 4·34-s − 7·37-s + 10·38-s + 4·39-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 3.02·7-s − 1.76·8-s + 1.20·11-s + 0.577·12-s + 1.10·13-s − 4.27·14-s + 5/4·16-s − 0.485·17-s − 1.14·19-s + 1.74·21-s − 1.70·22-s + 2.91·23-s − 1.02·24-s − 1.56·26-s + 3.02·28-s − 0.928·29-s − 1.25·31-s + 0.353·32-s + 0.696·33-s + 0.685·34-s − 1.15·37-s + 1.62·38-s + 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{4} \cdot 5^{8} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1883.32\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{825} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{4} \cdot 5^{8} \cdot 11^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.673108267\)
\(L(\frac12)\) \(\approx\) \(2.673108267\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_4$ \( 1 - T + T^{2} - T^{3} + T^{4} \)
5 \( 1 \)
11$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
good2$C_2^2:C_4$ \( 1 + p T + p T^{2} + 5 T^{3} + 11 T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \)
7$C_4\times C_2$ \( 1 - 8 T + 17 T^{2} + 50 T^{3} - 299 T^{4} + 50 p T^{5} + 17 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
13$C_2^2:C_4$ \( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 50 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \)
17$C_4\times C_2$ \( 1 + 2 T - 13 T^{2} - 60 T^{3} + 101 T^{4} - 60 p T^{5} - 13 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \)
19$C_4\times C_2$ \( 1 + 5 T + 6 T^{2} - 65 T^{3} - 439 T^{4} - 65 p T^{5} + 6 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
23$D_{4}$ \( ( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
29$C_2^2:C_4$ \( 1 + 5 T + 11 T^{2} + 195 T^{3} + 1736 T^{4} + 195 p T^{5} + 11 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \)
31$C_4\times C_2$ \( 1 + 7 T + 18 T^{2} - 91 T^{3} - 1195 T^{4} - 91 p T^{5} + 18 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
37$C_2^2:C_4$ \( 1 + 7 T + 32 T^{2} + 365 T^{3} + 3451 T^{4} + 365 p T^{5} + 32 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
41$C_2^2:C_4$ \( 1 + 17 T + 68 T^{2} - 661 T^{3} - 8425 T^{4} - 661 p T^{5} + 68 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
43$D_{4}$ \( ( 1 - 7 T + 67 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \)
47$C_2^2:C_4$ \( 1 - 13 T + p T^{2} - 325 T^{3} + 4016 T^{4} - 325 p T^{5} + p^{3} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \)
53$C_2^2:C_4$ \( 1 - 9 T - 7 T^{2} + 525 T^{3} - 3884 T^{4} + 525 p T^{5} - 7 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \)
59$C_4\times C_2$ \( 1 + 15 T + 76 T^{2} + 675 T^{3} + 8161 T^{4} + 675 p T^{5} + 76 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \)
61$C_4\times C_2$ \( 1 + 7 T - 12 T^{2} - 511 T^{3} - 2845 T^{4} - 511 p T^{5} - 12 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \)
67$D_{4}$ \( ( 1 + 21 T + 243 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \)
71$C_4\times C_2$ \( 1 - 8 T - 7 T^{2} + 624 T^{3} - 4495 T^{4} + 624 p T^{5} - 7 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
73$C_2^2:C_4$ \( 1 + T + 68 T^{2} + 155 T^{3} + 5911 T^{4} + 155 p T^{5} + 68 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \)
79$C_2^2:C_4$ \( 1 - 15 T + 21 T^{2} + 145 T^{3} + 2916 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \)
83$C_2^2:C_4$ \( 1 - 14 T + 53 T^{2} - 870 T^{3} + 14801 T^{4} - 870 p T^{5} + 53 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \)
89$D_{4}$ \( ( 1 - 15 T + 223 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \)
97$C_2^2:C_4$ \( 1 + 17 T + 12 T^{2} - 1445 T^{3} - 15649 T^{4} - 1445 p T^{5} + 12 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.42797023745197001597759160695, −7.41309031391923619252741329920, −7.07186795255288865495525918424, −6.64127293570398373611064455237, −6.37811955303380402735548935020, −6.37580130901964786007775547966, −6.17843350486515545370776951254, −5.59048646960539845377784768621, −5.49778363760124556416854223819, −5.22893066920209753709822214577, −5.11905164334217087234796688373, −4.56064536858319711260045220679, −4.52546932825995211053619351422, −4.25417018101278678568661365300, −3.96834881154801972310894742040, −3.66534184050959743334984742927, −3.27270605801694435473804817222, −3.01836473796484838681013901651, −2.77122121547077202295512794960, −2.08329293964844608379847082669, −2.06086446188747426896356905546, −1.81688313921970856044095769357, −1.39499216688172695756885602793, −0.971518130111110476105067919486, −0.58196919480990581816383050627, 0.58196919480990581816383050627, 0.971518130111110476105067919486, 1.39499216688172695756885602793, 1.81688313921970856044095769357, 2.06086446188747426896356905546, 2.08329293964844608379847082669, 2.77122121547077202295512794960, 3.01836473796484838681013901651, 3.27270605801694435473804817222, 3.66534184050959743334984742927, 3.96834881154801972310894742040, 4.25417018101278678568661365300, 4.52546932825995211053619351422, 4.56064536858319711260045220679, 5.11905164334217087234796688373, 5.22893066920209753709822214577, 5.49778363760124556416854223819, 5.59048646960539845377784768621, 6.17843350486515545370776951254, 6.37580130901964786007775547966, 6.37811955303380402735548935020, 6.64127293570398373611064455237, 7.07186795255288865495525918424, 7.41309031391923619252741329920, 7.42797023745197001597759160695

Graph of the $Z$-function along the critical line